Chapter 3.9, Problem 31E

Exponential model The following table shows the time of useful consciousness at various altitudes in the situation where a pressurized airplane suddenly loses pressure. The change in pressure drastically reduces available oxygen, and hypoxia sets in. The upper value of each time interval is roughly modeled by T = 10 · 2^−0.274a, where T measures time in minutes and a is the altitude over 22,000 in thousands of feet (a = 0 corresponds to 22,000 ft).

Altitude (in ft)	Time of Useful Consciousness
22,000	5 to 10 min
25,000	3 to 5 min
28,000	2.5 to 3 min
30,000	1 to 2 min
35,000	30 to 60 s
40,000	15 to 20 s
45,000	9 to 15 s

1. a. A Learjet flying at 38,000 ft (a = 16) suddenly loses pressure when the seal on a window fails. According to this model, how long do the pilot and passengers have to deploy oxygen masks before they become incapacitated?
2. b. What is the average rate of change of T with respect to a over the interval from 24,000 to 30,000 ft (include units)?
3. c. Find the instantaneous rate of change dT/da, compute it at 30,000 ft. and interpret its meaning.